Emergence in Complex Systems

This mini-program will focus on quantum materials, models and field theories where strong correlations result in novel emergent phenomena. The aim will be to highlight advances in the theory of magnetism, topological phases, superconductivity, entanglement, many-body localization, and other topics of recent interest.

The program will consist of organized talks in the morning, with the afternoons free for collaboration. Seminar rooms will be available in the afternoon for participants to organize their own informal discussions.

We demonstrate that the many-body localized phase is characterized by the existence of infinitely many local conservation laws. We argue that many-body eigenstates can be obtained from product states by a sequence of nearly local unitary transformation, and therefore have an area-law entanglement entropy, typical of ground states. Using this property, we construct the local integrals of motion in terms of projectors onto certain linear combinations of eigenstates [1]. The local integrals of motion can be viewed as effective quantum bits which have a conserved z-component that cannot decay. Thus, the dynamics is reduced to slow dephasing between distant effective bits. For initial product states, this leads to a characteristic slow power-law decay of local observables, which is measurable experimentally, as well as to logarithmic in time growth of entanglement entropy [2,3]. We support our findings by numerical simulations of random-field XXZ spin chains. Our work shows that the many-body localized phase is locally integrable, reveals a simple entanglement structure of eigenstates, and establishes the laws of dynamics in this phase.

In a conventional Mott insulator, magnitude of local spin moments remain fixed. They are `fixed spin Mott insulators'. We suggest that, in a multi orbital Hubbard model, when local Hund coupling is won over by inter-orbital superexchange couplings between neighboring sites, local spin moment can decrease its value in a cooperative fashion, through a first order phase transition, These are `Low spin state Mott insulators' (LSSMI). The minimal value of spins that can be reached are zero (half), for even (odd) number of electrons per site, We show that in LSSMI, depending on orbital degeneracy and electron number per site, novel quantum spin liquids can emerge, We discuss systems such as Fe arsenide, Fe selenide family and La2CoO3 in the light of our proposal. Certain long standing puzzles, including absence of any magnetic phase transition in La2CoO3 is explained in terms of a novel quantum spin liquid. Some properties of this spin liquid, a liquid of `quantum strings' will be discussed and some predictions made.

Standard picture of a topologically-nontrivial phase of matter is an insulator with a bulk energy gap, but metallic surface states, protected by the bulk gap. Recent work has shown, however, that certain gapless systems may also be topologically nontrivial, in a precise and experimentally observable way. In this talk I will review our work on a class of such systems, in which the nontrivial topological properties arise from the existence of nondegenerate point band-touching nodes (Weyl nodes) in their electronic structure. Weyl nodes generally exist in any three-dimensional material with a broken time-reversal or inversion symmetry. Their effect is particularly striking, however, when the nodes coincide with the Fermi energy and no other states at the Fermi energy exist. Such "Weyl semimetals" have vanishing bulk density of states, but have gapless metallic surface states with an open (unlike in a regular two-dimensional metal) Fermi surface ("Fermi arc"). I will discuss our proposal to realize Weyl semimetal state in a heterostructure, consisting of alternating layers of topological and ordinary insulator, doped with magnetic impurities. I will further show that, apart from Weyl semimetals, even such "ordinary" materials as common metallic ferromagnets, in fact also possess Weyl nodes in the electronic structure, leading to the appearance of chiral Fermi-arc surface states and the corresponding contribution to their intrinsic anomalous Hall conductivity.

Anushya Chandran, Perimeter Institute

How universal is the entanglement spectrum?

It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this talk, I will present evidence to the contrary. I will show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the ES and the Renyi entropies can mislead entirely, while for quantum Hall systems the ES has much less universal content than assumed to date. I will also discuss the consequences of the eigenstate thermalization hypothesis for the entanglement Hamiltonian, showing that a pure state in a sub-system can capture the properties of the reduced density matrix.

Paul Fendley, University of Virginia

Geometrical dependence of information in 2d critical systems

In both classical and quantum critical systems, universal contributions to the mutual information and Renyi entropy depend on geometry. I will first explain how in 2d classical critical systems on a rectangle, the mutual information depends on the central charge in a fashion making its numerical extraction easy, as in 1d quantum systems. I then describe analogous results for 2d quantum critical systems. Specifically, in special 2d quantum systems such as quantum dimer/Lifshitz models, the leading geometry-dependent term in the Renyi entropies can be computed exactly. In more common 2d quantum systems, numerical computations of a corner term hint toward the existence of a universal quantity providing a measure of the number of degrees of freedom analogous to the central charge.

Matthew Fisher, University of California, Santa Barbara

Colloquium: Quantum Tapestries

Within each of nature's crystals is an exotic quantum world of electrons weaving to and fro. Each crystal has it's own unique tapestry, as varied as the crystals themselves. In some crystals, the electrons weave an orderly quilt. Within others, the electrons are seemingly entwined in an entangled web of quantum motion. In this talk, I will describe the ongoing efforts to disentangle even nature's most intricate quantum embroidery. Cutting-edge quantum many-body simulations together with recent ideas from quantum information theory, such as entanglement entropy, are enabling a coherent picture to emerge.

In this talk, I will show the emergence of p+ip topological superconducting ground state in infinite-U Hubbard model on honeycomb lattice, from both state-of-art Grassmann tensor-network numerical approach and quantum field theory approach.

Duncan Haldane, Princeton University

Geometry of topological matter: some examples

I will look at two cases of the interplay of geometry (curvature) and topology:(1) 3D Topological metals: how to understand their surface "Fermi arcs" in terms of their emergent conservation laws and the Streda formula for the non-quantized anomalousHall effect. (2) The Hall viscosity tensor in the FQHE as a local field, and its Gaussian-curvature response that allows local compression or expansion of the fluid to accommodate substrate inhomogeneity

David Hawthorn, University of Waterloo

Charge density wave order in cuprate superconductors

Despite over 25 years of intensive research, the problem of cuprate high temperature superconductors remains unsolved. One aspect that makes understanding the cuprates difficult is that the superconducting phase competes with other phases of matter, in particular a charge density wave (CDW) state – a spontaneous, spatial modulation of the charge density. It is now clear that understanding the CDW state is of central importance to understanding many aspects of the cuprates. Recently, breakthroughs in the investigation of CDW order and superconductivity in the cuprates have occurred using a novel experimental technique, resonant soft x-ray scattering. In this talk, I will discuss experimental investigations of the CDW in the cuprates using resonant x-ray scattering. These experiments provide key insights into the doping phase diagram of CDW order, the microscopic character of the CDW order and the nature of the competition between CDW and superconducting order parameters, which we argue are intertwined in a multi-dimensional order parameter that experiences thermal angular fluctuations in the pseudogap phase.

Some time ago (1999), Dy2Ti2O7, was shown to be a magnetic analog of water ice, and thus dubbed "spin ice". Recently, theories and experiments have developed the perspective of viewing excitations within the low temperature phase of this spin ice as monopoles. I will present early results of specific heat, ac susceptibility and magnetization measurements as well as my group's recent results on this system

Chris Laumann, Princeton University

Many-body localization with dipoles

Statistical mechanics is the framework that connects thermodynamics to the microscopic world. It hinges on the assumption of equilibration; when equilibration fails, so does much of our understanding. In isolated quantum systems, this breakdown is captured by the phenomenon known as many-body localization. This breakdown manifests in a variety of ways, as elucidated by much recent theoretical and numerical work. Many-body localized phases violate Ohm's law and Fourier's law as they conduct neither charge nor heat; they can exhibit symmetry breaking and/or topological orders in dimensions normally forbidden by Mermin-Wagner arguments; they hold potential as strongly interacting quantum computers due to the slow decay of local coherence.In this talk, I will briefly introduce the basic phenomena of many-body localization and review its theoretical status. To date, none of these phenomena has been observed in an experimental system, in part because of the isolation required to avoid thermalization. I will consider several dipolar systems which we believe to be ideal platforms for the realization of MBL phases and for investigating the associated delocalization phase transition. The presence of strong interactions in these systems underlies their potential for exploring physics beyond that of single particle Anderson localization. However, the power law of the dipolar interaction immediately raises the question: can localization in real space persist in the presence of such long-range interactions?I will review and extend several arguments producing criteria for localization in the presence of power laws and present small-scale numerics regarding the MBL transition in several of the proposed dipolar systems.Associated preprint: N. Yao, CRL, S. Gopalakrishnan, M. Knap, M. Mueller, E. Demler., M. Lukin arXiv:1311.7151

Yuan-Ming Lu, University of California, Berkeley

A unification of symmetric Z2 spin liquids on kagome lattice

While there is mounting numerical evidence for a gapped Z2 spin liquid in the kagome Heisenberg model, a complete characterization of this topological phase remains to be accomplished. A defining property, the projective symmetry group (PSG) which fixes how the emergent excitations of the spin liquid phase transform under symmetry, remains to be determined. Following a Chern-Simons field theory, we show how PSG determines measurable properties of a Z2 spin liquid, such as the existence of symmetry protected gapless edge states. This fact enables us to unify two distinct types of projected wavefunctions for Z2 spin liquids: the Schwinger-boson states and the fermionic spinon states. We also provide concrete predictions for identifying the spin liquid ground state on the kagome lattice.

A 3d electron topological insulator (ETI) is a phase of matter protected by particle-number conservation and time-reversal symmetry. It was previously believed that the surface of an ETI must be gapless unless one of these symmetries is broken. A well-known symmetry-preserving, gapless surface termination of an ETI supports an odd number of Dirac cones. In this talk, I will show that in the presence of strong interactions, an ETI surface can actually be gapped and symmetry preserving, at the cost of carrying an intrinsic two-dimensional topological order. I will argue that such a topologically ordered phase can be obtained from the surface superconductor by proliferating the flux 2hc/e vortex. The resulting topological order consists of two sectors: a Moore-Read sector, which supports non-Abelian charge e/4 anyons, and an Abelian anti-semion sector, which is electrically neutral. The time-reversal and particle number symmetries are realized in this surface phase in an "anomalous" way: one which is impossible in a strictly 2d system. If time permits, I will discuss related results on topologically ordered surface phases of 3d topological superconductors.

Are non-Fermi-liquids stable to pairing?

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles are expected to arise in a number of physical systems. Examples include i) quantum critical points associated with the onset of order in metals, ii) the spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator and iii) the Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this talk, I will use renormalization group techniques to investigate possible instabilities of such non-Fermi-liquids to pairing. I will show that for a large class of phase transitions in metals, the attractive interaction mediated by order parameter fluctuations always leads to a superconducting instability, which preempts the non-Fermi-liquid effects. On the other hand, the spinon Fermi-surface and the Halperin-Lee-Read states are stable against pairing for a sufficiently weak attractive short-range interaction. However, once the strength of attraction exceeds a critical value, pairing sets in. I will describe the ensuing quantum phase transition between i) the U(1) and the Z_2 spin-liquid states, and ii) the Halperin-Lee-Read and Moore-Read states.

Zlatko Papic, Perimeter Institute

Topological phases in graphene

As realized for the first time in 1980s, quantum many-body systems in reduced spatial dimensions can sometimes undergo a special type of ordering which does not break any symmetry but introduces long-range entanglement and emergent excitations that have radically different properties from their original constituents. Most of our experimental knowledge of such ``topological" phases of matter comes from studies of two-dimensional electron gases in GaAs semiconductors in high magnetic fields and at low temperatures. In the first part of this talk, I will give an introduction to these systems and review some latest theoretical developments related to their entanglement properties. In the second part, I will discuss new possibilities for experimental realizations of topological phases in bilayer graphene. I will present evidence that this material supports an ``even-denominator" fractional state, related to the Moore-Read state, whose observation has recently been reported. Finally, I will outline several proposals based on the tunability of the electron-electron interactions in bilayer graphene which might enable further experimental progress beyond GaAs.

We show that double perovskites with 3d and 5d transition metal ions exhibit spin-orbit coupled magnetic excitations, finding good agreement with neutron scattering experiments in bulk powder samples. Motivated by experimental developments in the field of oxide heterostructures, we also study double perovskites films grown along the [111] direction. We show that spin-orbit coupling in such low dimensional systems can drive ferromagnetic order due to electronic correlations. This results in topological Chern bands, with symmetry-allowed trigonal deformations leading to quantum anomalous Hall states supporting a pair of chiral edge modes.

Luiz Santos, Perimeter Institute

Aharonov-Bohm effect in symmetry protected states

Symmetry protected topological (SPT) states are generalizations of topological band insulators to interacting systems. They possess a gapped bulk spectrum together with symmetry protected edge states, with no topological order. There has been recently an intense effort to classify SPT states both in terms of group cohomology as well as from the point of view of effective field theories. An interesting related question is to understand the structute of lattice models that realize SPT physics. In this talk, I shall present a class of lattice models describing the egde of non-chiral two-dimensional bosonic SPT states protected by Z_N symmetry. A crucial aspect of the construction relies on finding the correct non-trivial Z_N symmetry realizations on the edge consistent with all the possible classes of SPT states. Then I shall discuss the Aharonov-Bohm effect on the many-body SPT state by studying this many-body effect on the aforementioned gapless edge states. The effect of a Z_N gauge flux on the egde states is formulated in terms of twisted boundary conditions of the lattice models. The low energy spectral shifts due to the gauge flux are shown to depend on each of the SPT classes in a predictable way. I shall, in the course of this talk, present numerical results of exact diagonalization of our lattice Hamiltonians that support this analysis. This work is done in collaboration with Juven Wang and appears in arXiv:1310.8291.

Brian Swingle, Harvard University

Incoherent metals

I'll talk about some work in progress concerning the topic of metals which have no coherent quasiparticles. In particular, I'll compare and contrast the ubiquitous near horizon AdS2 region appearing in holographic models with a phase of matter called the spin incoherent luttinger liquid. By analyzing the structure of entanglement and correlations, we will find many similarities between these two states of matter. An interpretation of some incoherent metals as describing intermediate scale renormalization group fixed poins with an infinite number of relevant directions will also be discussed.

Senthil Todadri, Massachusetts Institute of Technology

Interacting electronic topological insulators in three dimensions

I will review recent progress in describing interacting electronic topological insulators/superconductors in three dimensions. The focus will be on Symmetry Protected Topological (SPT) phases of electronic systems with charge conservation and time reversal. I will argue that the well known Z2 classification of free fermion insulators with this important symmetry generalizes to a Z2^3 classification in the presence of interactions. I will describe the experimental fingerprints and other physical properties of these states. If time permits, I will describe results on the classification and properties of 3d electronic SPT states with various other physically relevant symmetries.

Topological phases in frustrated quantum spin systems have fascinated researchers for decades. One of the earliest proposals for such a phase was the chiral spin liquid put forward by Kalmeyer and Laughlin in 1987 as the bosonic analogue of the fractional quantum Hall effect. Elusive for many years, recent times have finally seen a number of models that realize this phase. However, these models are somewhat artificial and unlikely to be found in realistic materials.Here, we take an important step towards the goal of finding a chiral spin liquid in nature by examining a physically motivated model for a Mott insulator on the Kagome lattice with broken time-reversal symmetry. We first provide a theoretical justification for the emergent chiral spin liquid phase in terms of a network model perspective. We then present an unambiguous numerical identification and characterization of the universal topological properties of the phase, including ground state degeneracy, edge physics, and anyonic bulk excitations, by using a variety of powerful numerical probes, including the entanglement spectrum and modular transformations.

Zhenghan Wang, Microsoft Station Q

Gauging symmetry of 2D topological phases

I will discuss the mathematical framework for gauging a local unitary finite group symmetry of a 2D topological phase of matter.

Pavel Wiegmann, University of Chicago

Fractional Quantum Hall Effect in a curved space

We developed a general method to compute the correlation functions of FQH states on a curved space. The computation features the gravitational trace anomaly and reveals geometric properties of FQHE. Also we highlight a relation between the gravitational and electromagnetic responce functions. The talk is based on the recent paper with T. Can and M. Laskin.

Statistical mechanics is the framework that connects thermodynamics to the microscopic world. It hinges on the assumption of equilibration; when equilibration fails, so does much of our understanding. In isolated quantum systems, this breakdown is captured by the phenomenon known as many-body localization. This breakdown manifests in a variety of ways, as elucidated by much recent theoretical and numerical work.

In a conventional Mott insulator, magnitude of local spin moments remain fixed. They are `fixed spin Mott insulators'. We suggest that, in a multi orbital Hubbard model, when local Hund coupling is won over by inter-orbital superexchange couplings between neighboring sites, local spin moment can decrease its value in a cooperative fashion, through a first order phase transition, These are `Low spin state Mott insulators' (LSSMI).

While there is mounting numerical evidence for a gapped Z2 spin liquid in the kagome Heisenberg model, a complete characterization of this topological phase remains to be accomplished. A defining property, the projective symmetry group (PSG) which fixes how the emergent excitations of the spin liquid phase transform under symmetry, remains to be determined. Following a Chern-Simons field theory, we show how PSG determines measurable properties of a Z2 spin liquid, such as the existence of symmetry protected gapless edge states.

It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this talk, I will present evidence to the contrary. I will show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low energy spectrum exhibit singular changes, even when the physical system remains in the same phase.

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles are expected to arise in a number of physical systems. Examples include i) quantum critical points associated with the onset of order in metals, ii) the spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator and iii) the Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this talk, I will use renormalization group techniques to investigate

I'll talk about some work in progress concerning the topic of metals which have no coherent quasiparticles. In particular, I'll compare and contrast the ubiquitous near horizon AdS2 region appearing in holographic models with a phase of matter called the spin incoherent luttinger liquid. By analyzing the structure of entanglement and correlations, we will find many similarities between these two states of matter.

I will review recent progress in describing interacting electronic topological insulators/superconductors in three dimensions. The focus will be on Symmetry Protected Topological (SPT) phases of electronic systems with charge conservation and time reversal. I will argue that the well known Z2 classification of free fermion insulators with this important symmetry generalizes to a Z2^3 classification in the presence of interactions. I will describe the experimental fingerprints and other physical properties of these states.

Topological phases in frustrated quantum spin systems have fascinated researchers for decades. One of the earliest proposals for such a phase was the chiral spin liquid put forward by Kalmeyer and Laughlin in 1987 as the bosonic analogue of the fractional quantum Hall effect. Elusive for many years, recent times have finally seen a number of models that realize this phase. However, these models are somewhat artificial and unlikely to be found in realistic materials.

Within each of nature's crystals is an exotic quantum world of electrons weaving to and fro. Each crystal has it's own unique tapestry, as varied as the crystals themselves. In some crystals, the electrons weave an orderly quilt. Within others, the electrons are seemingly entwined in an entangled web of quantum motion. In this talk, I will describe the ongoing efforts to disentangle even nature's most intricate quantum embroidery.